I did a biology experiment where I increased the temperature of the incubator for incubation of E. coli, where I then increased the temperature above the suitable temperature range. I measured the E. coli growth rate. Basically there was a positive exponential growth rate, even when increased above the temperature range.

I'm thinking about using Spearman Correlation Coefficient to prove that the exponential behaviour of the E. coli growth (even above E. coli's suitable temperature range) is caused by the temperature increase.

Is it suitable to use Spearman Correlation Coefficient in this situation?

I applied it to the formula and got a regression value of 0.82. Understandably, since the increase in temperature gradually cause a decrease growth rate. Thanks All

First, if you are going to do a regression, you also want a p-value along with the regression value. It is much easier to get a very high regression value with a few data points by accident

Second, the Spearman Correlation only determines whether two values monotomically increase together. r=0.82 just indicates that growth rate tends to increase as temperature does. It doesn't tell you whether the growth is exponential, cubic, quadratic, linear, or something else. If you are trying to show that the growth rate follows a specific function, you need a correlation coefficient tuned to that specific function. In the case of linear correlation, this would be the Pearson Correlation. If you believe the correlation is exponential, then the log of the growth rate should be linear with the temperature. So you can just take the log of the growth and run a Pearson Correlation. If you don't get a strong regression value, this means that the two values increase together, but not exponentially.

Third, I would suggest being a tad bit more specific. From what I am reading from your post, what you are saying is that (outside of the suitable range) the growth starts as exponential, tapers off, and then decreases. In this case trying to run a correlation on the entire range and claiming that is non-stop growth would simply be wrong. What you would want is to break the data down into the regions where growth rate increases, where growth rate is constant, and where growth rate decreases, and run correlations on each region separately. Assuming you have enough data points for that at least.